1511 = (2 * 755) + 1 ;
hence not 2 divides 1511 by NAT_4:9; :: thesis:
1511 = (3 * 503) + 2 ;
hence not 3 divides 1511 by NAT_4:9; :: thesis:
1511 = (5 * 302) + 1 ;
hence not 5 divides 1511 by NAT_4:9; :: thesis:
1511 = (7 * 215) + 6 ;
hence not 7 divides 1511 by NAT_4:9; :: thesis:
1511 = (11 * 137) + 4 ;
hence not 11 divides 1511 by NAT_4:9; :: thesis:
1511 = (13 * 116) + 3 ;
hence not 13 divides 1511 by NAT_4:9; :: thesis:
1511 = (17 * 88) + 15 ;
hence not 17 divides 1511 by NAT_4:9; :: thesis:
1511 = (19 * 79) + 10 ;
hence not 19 divides 1511 by NAT_4:9; :: thesis:
1511 = (23 * 65) + 16 ;
hence not 23 divides 1511 by NAT_4:9; :: thesis:
1511 = (29 * 52) + 3 ;
hence not 29 divides 1511 by NAT_4:9; :: thesis:
1511 = (31 * 48) + 23 ;
hence not 31 divides 1511 by NAT_4:9; :: thesis:
1511 = (37 * 40) + 31 ;
hence not 37 divides 1511 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1511 & n is prime holds
not n divides 1511 by XPRIMET1:24;
hence 1511 is prime by NAT_4:14; :: thesis: